Thoughts on Hierarchical Modeling Methods for Complex ...3)_2009_419-430.pdf · Thoughts on Hierarchical Modeling Methods for Complex Structures David W. Rosen Georgia Institute of

Computer-Aided Design & Applications, 6(3), 2009, 419-430

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Computer-Aided Design and Applications© 2009 CAD Solutions, LLC

http://www.cadanda.com

Thoughts on Hierarchical Modeling Methods for Complex Structures

David W. Rosen

Georgia Institute of Technology, [email protected]

ABSTRACT

Hierarchical modeling refers to the modeling of parts with solid geometry, materialcomposition, and possibly distributions of properties at multiple scales. Typical solidand heterogeneous models represent geometry at one scale but do not containgeometric information about features at scales that are orders of magnitude smaller.In this paper, the related topics of heterogeneous and hybrid modeling, subdivisionmethods, and level of detail approaches are surveyed. The capabilities of thesetechnologies are compared against the requirements for modeling cellular materialsand other structures with complex geometric and material constructions. A specifichierarchical modeling method is proposed. Three 2D examples are used to illustratethe application of the proposed modeling method.

Keywords: hierarchical modeling, heterogeneous modeling, solid modeling.DOI: 10.3722/cadaps.2009.419-430

1. INTRODUCTIONManufactured parts typically have a complex distribution of microstructure, mechanical properties,and sometimes materials. Commercial CAD systems are not able to represent such distributions, butsome heterogeneous or multi-material research systems are able to represent at least some of thisinformation. In this work, we are interested in modeling methods that enable the capture of geometryas well as material and microstructure distributions at multiple length scales. This requirement formultiple length scales leads to the term “hierarchical modeling” and is meant to include geometry,material, and microstructure distributions. That is, hierarchical modeling implies heterogeneousmodeling, at least in this work.

The specific engineering domain of interest is that of cellular materials and other parts fabricatedusing additive manufacturing (AM) (i.e., “rapid prototyping”) processes. However, the domain ofapplication of hierarchical modeling is much broader. The concept of cellular materials is motivatedby the desire to put material only where it is needed for a specific application. Achieving high stiffnessor strength and minimal weight are typical objectives [13, 23]. Multifunctional requirements are alsocommon, such as structural strength and vibration absorption. We hypothesize that designedmesostructures will enable structures and mechanisms to be designed that perform better than partswith bulk or non-designed mesostructures, particularly for multifunctional applications. Testing thishypothesis requires the ability to bridge multiple length scales (micro to macro, or even nano tomacro). Fig. 1 shows several images of cellular structures and a cross-section of a metal partfabricated using a laser cladding process (LENS), which illustrates some aspects of typicalmicrostructures that result from laser AM processes.


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To achieve hierarchical modeling, it is first helpful to identify specific requirements on heterogeneousand multi-scale methods. The requirements that we propose include the capability to:

Represent and design with hundreds of thousands of shape elements, enabling large complexdesign problems as well as designed material mesostructures.

Represent complex material compositions and microstructures and ensure that they arephysically meaningful.

Determine mechanical properties from material compositions and mesostructures acrosslength scales.

Ensure that specified shapes, material structures, and properties are manufacturable. Ensure that as-manufactured designs achieve requirements.

Of these requirements, the first two are addressed directly in this paper. The other three highlightimportant properties of the modeling method for supporting other design and manufacturingobjectives.

a) extruded shapes b) 3D lattice structure c) microstructure from LENSprocess

Fig. 1: Example cellular structures and additive manufacturing microstructure.

A Design for Additive Manufacturing (DFAM) framework was introduced in some of our earlier work[23] and is illustrates our desire to achieve design requirements by manipulating both the geometry ofa part and its material composition. This framework is based upon the process-structure-propertyrelationships from the materials science domain [19], where analysis of a material consists ofexamining the microstructure of the material after processing it, and determining its mechanicalproperties from the microstructure. In materials design, material developers seek to reverse theprocess by specifying desired behavior, deriving target mechanical properties, designing desiredmicrostructures, and developing manufacturing process conditions to achieve those microstructures.

In the design of cellular materials, we introduce an additional hierarchy into this framework thatcaptures the geometric complexity of the microstructure or material composition. As a result, partbehaviors can be controlled by adjusting cellular structure geometry in addition to materialmicrostructure. The concept is illustrated in Fig. 2. Let a solid model with material composition and

other material-related attributes be denoted as = (, ), where is the solid geometric model and

is the attribute space, with some attributes being applied only to regions of . denotes the structure

of the design. The manufacturing process space, , consists of process plans with sequences of

operations and values of process variables. Property space contains information about part

properties that are derivable from S using physical principles; e.g., mechanical, thermal, and electrical

properties. Finally, we will add behavior space, , which contains information about a part’s actual and

desired behavior given some loading and boundary conditions.

The overall DFAM method consists of a traversal of the framework in Fig. 2 from Behavior to Process,then back again to Behavior. The traversal from Behavior to Process can be called design, wherefunctional requirements are mapped to properties and geometry that satisfy those requirements tostructures and through process planning to arrive at a potential manufacturing process. Going in the


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reverse direction, one can simulate the designed device and its manufacturing process to determinehow well it satisfies the original requirements.

Material

PG

PM

ProcessTM

PropertySM

Structure

SG B Behavior

PM

ProcessSM

Structure

SG TG

Geometry

Decreasing size scale


Material

PGPG

PM

ProcessTM

PropertySM

Structure

SG

SM

Structure

SG B BehaviorB Behavior

PM

ProcessSM

Structure

SG

SM

Structure

SG TGTG

Geometry



Fig. 2: DFAM framework with geometry and material layers.

In this paper, the area of heterogeneous modeling is surveyed briefly, then a more specific descriptionof multiresolution modeling method is offered in order to identify methods and models from thegeometric modeling literature that may contribute to hierarchical modeling. From these surveys, anew hierarchical modeling method is proposed in Section 4. Three simple 2-dimensional examples arepresented in Section 5, one that illustrates multi-scale geometric modeling, one that demonstratesmulti-scale material modeling, and one that demonstrates randomized fiber orientations that varywith variations in geometry. Conclusions are offered in the final section.

2. HETEROGENEOUS MODELINGBroadly speaking, the objective of heterogeneous modeling is to model a part’s geometry and thedistribution of materials and other properties throughout the geometry. Many heterogeneousmodeling methods have been developed by different research groups. Most of this work can beclassified broadly into two categories: spatial discretization methods, and implicit and other non-discretization methods. Research on discretization methods started with the observation that materialcompositions could be assigned to specific volumes within a part. Finite element modeling approachesassigned material compositions to individual elements or nodes; material compositions withinelements were interpolated in a manner similar to stress interpolation using shape functions [14, 15.].Rather than rely on finite element decompositions, other researchers applied voxel-basedrepresentations that utilized spatial occupancy enumeration of part geometry. Again, materialcomposition information was applied to either individual voxels or interpolated over sets of voxelsusing a part’s bounding surface [25, 29]. General cellular decompositions have also receivedconsiderable attention. Dutta and coworkers [3, 15, 16] have developed a series of heterogeneous solidmodeling approaches over several years. Material compositions are assigned to primitive geometricentities and a set of modeling operations was defined that operate on these entities.In the area of non-discretized approaches, some researchers have separated the representation ofmaterial compositions and properties from the underlying part geometry [27]. Others have utilizedimplicit modeling approaches, which has advantages in that a common mathematical model is used forboth geometry and material composition [12, 24]. A method based on hypertextures was proposed byPark et al. [20] and extended [21] that provides more intuitive user controls, according to thedevelopers.Probably the most sophisticated heterogeneous object modeling approach, due to Kumar et al. [15], isbased on fiber bundles. A fiber bundle model includes not only a geometric model of the part, but alsoa mapping from points located within the part to additional properties such as material composition.Thus geometry is considered to be the “base attribute,” and each point in the object is considered tocorrespond to a point in the base (Euclidean) space. The object geometry is represented as adescription of the overall geometry together with a finite set of disjoint decompositions which will beused to map into the attribute space.A typical model of geometry and material composition will be presented. A volume fraction-based n-phase, graded material model will be used here. The material composition at a point is represented byvolume fractions of n

mphases: (v

1, v

2, ..., v

nm). Then, the space of material compositions is given by Eqn.

1:


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m v1,...,v

nm

Rn

m | vi 1

i1

nm

(1)

The condition ensures that material combinations at all points are feasible (volume fractions sum to 1).For example, a combination of three materials could be given as (0.4, 0.5, 0.1), which indicates that thegeometric domain is composed of 40% material 1, 50% material 2, and 10% material 3.For convenience, we will introduce a set of pure materials, denoted by the standard basis in n-space;i.e., m

1= (1, 0, 0), m

2= (0, 1, 0), m

3= (0, 0, 1) if three materials are being modeled (n = 3). We will

denote this standard basis of materials by Mn. Then, a particular material combination will be given as{v

1∙m

1, v

2∙m

2, v

3∙m

3}, where again the v

irepresent volume fractions. The space of material compositions

can be rewritten as

VMn vm,m Mn ,v v1,...,v

nm

Rn

m | vi 1

i1

nm

(2)

Heterogeneous representations are often represented by the combination of geometry models, from a

space of geometric entities , and material models. As such, the space of heterogeneous models is

given by:

= × (3)

3. MULTI-RESOLUTION MODELINGMulti-resolution modeling methods are used typically formodel simplification or for level-of-detail control forvisualization or virtual reality applications. Multi-resolutionmodeling originated from the desire for computationalefficiency in computer graphics from the mid-1970’s, whenJames Clark [10] introduced hierarchically structured objectrepresentations, with progressively more detailedrepresentations for viewing objects that are increasingly closerto the viewer. An example used to motivate this work is thereplacement of a complex model with a coarse tessellation (e.g.,replace a sphere with a dodecahedron) if the viewer is far fromthe sphere, which is an example of level-of-detail control. Thefirst three steps of Catmull-Clark subdivision of a cube isshown in Fig. 3. In the limit, the subdivision converges to asphere. This concept was extended to objects modeled withparametric surfaces [6], which gave rise to the subdivisionsurface and volume methods. Broadly speaking, literature inthis area can be categorized based on subdivision or discrete geometric modeling approaches.

3.1 Subdivision ApproachesSubdivision approaches to multiresolution modeling are appealing in that they enable the addition ofshape details or smoothing by subdividing an appropriate mesh. Some of the simplest subdivisionapproaches work with a control polygon (curve) or polyhedron (surface) as the mesh is subdivided.Different approaches worked with triangular meshes or quadrilateral meshes [6]. Other approachesuse simplicial or other complexes for their mesh. From a design perspective, an interesting applicationof subdivision methods is to add shape detail at a particular subdivision step by manipulating themesh or control vertices. If smaller shape details are desired, one or more subdivision steps can beperformed, then the user can manipulate the mesh at the resulting level.As stated, the mesh can represent a control polygon or polyhedron. Tensor-based Bezier, B-spline, orNURBS parametric curve/surface formulations are often used [2]. That is, each face of a subdivision(see Fig. 3) is modeled by a parametric surface; vertices of the mesh are corner points of the surface’scontrol polyhedron. Others have taken a non-tensor-based approach, utilizing simplicial meshes andmultivariate splines [7], arguing that non-tensor-based methods are needed for non-structureddomains and they can achieve comparable smoothness with relatively low polynomial degree.

Fig. 3: First three steps of Catmull–Clark subdivision of a cube [28].


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Subdivision approaches have been extended to deal with volumes, enabling multiresolution modelingof 3D solids, not just their bounding curves and surfaces.A line of research has explored the integration of non-manifold and manifold regions [11, 30]. Changand Qin [8] presented a unified approach to non-manifold subdivision modeling, using a simplicialcomplex for their domain and box splines for interpolating surfaces. Subdivision rules are modified byspatial averaging to achieve smoothness conditions. They developed special subdivision rules forhandling regions between manifold and non-manifold topologies.The usage of subdivision approaches for modeling cellular structure is of questionable value since themain objective is to generate smooth surfaces and/or shape detail. However, the decomposition ofsimplicial complexes can model finer and finer partitions of part geometry, which may be useful inmodeling composite materials and possibly lattice structures. The interpolation of shapes usingsplines may be extended to modeling distributions of material compositions, particles or fibers forcomposite materials, and grains in metals.

3.2 Discrete Geometric ModelsTypically, discrete geometry refers to tessellated surfaces, enumeration methods for decomposingsolids (e.g., octrees), and cellular decompositions (e.g., simplicial complexes). Part models consist ofdiscrete vertices, edges, faces, tetrahedra, etc. A class of methods known as level-of-detail (LOD)techniques have been developed for visualization and virtual reality applications [5, 31]. Multiple partrepresentations are used, at different levels of detail, to minimize rendering times when parts are faraway from the viewer, while enabling accurate rendering when the viewer is close. The subdivisionmethods are used often for generating these representations, an example of which is shown in Fig. 3.Recently, other discrete representations have been developed with the objective of providing efficientmethods of constructing solid models, visualizing large models, or generating models for otherapplications such as analysis or manufacturing. In a variation of the octree representation, a discreterepresentation can be generated by classifying, against a part model, a set of sample points distributedon the nodes of a regular, axis-aligned grid [5]. Nodes lying inside the part model are colored black(for instance), while nodes outside the model are colored white. The part boundary can be representedas a collection of sticks, which are grid edges that connect black and white nodes.Chen and Wang [9] introduced the layered depth-normal images (LDNI) which can be used withgraphics hardware to very quickly compute Boolean operations between solid models. The LDNIrepresentation consists of three images of the part, one image along each coordinate axis. A ray fromeach pixel is projected into the part space to intersect with the part model at point p

ij, for the i,j pixel.

Associated with each pixel of each image is the depth of the surface from the viewing plane and thesurface normal at p

ij. Others have presented similar work, based on sampled rays, including ray-rep

[17] and Layered Depth Images (LDI) [26], without surface normal information. Some applicationsincluded solid model construction, collision detection, and visualization.One issue associated with the use of sampled discrete models is the need to convert from boundaryrepresentations to the discrete representation, and then back again for visualization (and othermodeling) purposes. Isosurface extraction algorithms are needed to reconstruct part surfaces from adiscrete representation [5]. Given a set of sampled points and/or depths, many ambiguities arise whentrying to reconstruct surfaces, such as locations of surface boundaries, surface continuities, sharpcorners, and the like. Marching cubes algorithms and their variants [18] are often used to computesurfaces, and various heuristics are often applied to resolve ambiguities [1].It is not clear if the types of discrete modeling approaches described here will benefit hierarchicalmodeling of cellular structures. These approaches may be helpful to increase efficiency for variouscomputations, but probably will not serve as the primary representation for composite materials orlattice structures.

3.3 Hybrid ApproachesA variety of other approaches have been investigated. One approach integrated precise solid modelinginto a virtual reality (VR) environment [31]. The representation consisted of six levels of information:assemblies, parts, features, feature elements, geometric and topological relationships, and polygons.These six levels were organized into three layers, including a high layer constraint-based model forprecise object definition, a mid-layer constructive solid geometry/boundary representation hybridsolid model for supporting the hierarchically geometry abstractions and object creation, and a low


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layer polygon model for real-time visualization and interaction in the VR environment. The constraintsystem was used to support object creation and also to support VR interactions.Another approach sought to integrate NURBS-based design, analysis, and optimization using ahierarchical partition of unity construction [22]. This approach shares some similarities with thesubdivision approaches. It also integrates consideration of material distributions within part models,so is a heterogeneous representation. In their method, NURBS are used to discretize the geometry,material, and behavioral fields in their representation. Behavioral fields are associated with thegeometric primitives and their values are determined by a global analysis problem. The authorsshowed that this NURBS construction leads to recursive partition of unit constructions on the knotspaces due to the partition of unity properties of NURBS basis functions. Examples demonstratedapplications to simultaneous topology and material distribution optimization, shape and sizeoptimization, and optimal design to mitigate the effects of cracks.

4. HIERARCHICAL MODELA hierarchical modeling approach is proposed that enables the specification of geometric and materialcomposition models at multiple size scales. The model is based on the concepts of subdivision andheterogeneous modeling methods introduced earlier. The approach is also based on concepts oftextured surfaces and solids.

4.1 Hierarchical Modeling ConceptsThe proposed hierarchical modeling approach is to support geometric and material models at multiplesize scales. A part model can be given as a 3-D geometric model and a material composition at themacro scale. In addition, the designer can specify a more precise geometry and material compositionas smaller size scales. For example, a part may have a shape appropriate for a product housing and isto consist of nylon with 10% by volume of a nano-clay for stiffening. At the macro scale, the nano-clayfiller (say, with mean diameter 500 nm) is too small to represent exactly in the geometric model.However, the nano-clay geometry becomes important when considering the part microstructure. Inthis case, the micro-scale geometry becomes important and the material composition cannot beassumed to be uniformly distributed.

In heterogeneous modeling, a part model contains a geometric model and a material model. Typically,

the part modeling space, , is given by the product of a space of 3-D geometric models, , and a

material composition space, (recall Eqn. 3). In the proposed hierarchical modeling approach, part

models will be composed of a macro scale part model, M, one or more smaller scale models, denoted

for micro-scale or n

for nano-scale, and a set of rules (RulesM

, Rulesmn

) that relate the different size

scales, as given in Eqn. 1:

MRules

M

Rules

n

n(4)

4.2 Hierarchical Geometric ModelsWe will borrow methods from the subdivision literature (e.g., [2, 6, 8]) in order to develop rules forrelating size scales. Curves, surface, and solids can be subdivided using different methods. In general,any subdivision method can be used to divide part geometry into smaller regions. When geometricsubdivisions become appropriately small, the geometry for the smaller size scale is inserted into eachsubdivision. The material composition models are handled similarly. Fig. 4 shows an example with ahollow diamond microstructure (unit size) that is mapped into a subdivided region in a larger 2Dshape. In the microstructure, the solid region is all of a given material, while the white regions arevoids. The macro-scale geometric domain is composed of the same material as the microstructure. In

this case, gM

Mis the macro-scale geometry, {gs

, gv} is the microstructure geometry (gs

, gv ),

where gsrepresents solid and gv

represents void, the two materials are m1

= (1, 0), m2

= (0, 1)

(representing solid and void, respectively), mM

Mis m

1, and m is {m

1, m

2}. As a result, s

M=

(gM, m

M), s

M

M, s = {( gs

, m1), (gv

, m2)}, s , s

n= , and rules

M= {subdivision by mapping, relate

size scales via linear interpolation}.


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In this paper, we will use a simple mapped mesh subdivision approach, partitioning surface patchesinto smaller patches (quadrilateral subdivision) and partitioning solid volumes into smaller hexahedralvolumes. Geometry from the micro-scale model is then mapped into each subdivided patch or volumewith the appropriate material composition.We can go a step further and generate intermediate models between given size scales. The approachtaken in this paper is to map geometry from the smaller scale into regions generated at eachsubdivision step. Material compositions are interpolated from those given at the larger and smallerscales. B-spline models are used to interpolate material compositions across the subdivision steps.This enables the use of linear or quadratic interpolation across many subdivisions. Resultingintermediate models will share some of the shape and material composition characteristics of thelarger and smaller scale models and may be suitable for visualization or analysis at intermediate sizescales.

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m]

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Fig. 4: Mapping a unit geometric primitive into subdivided region.

4.3 Hierarchical Material ModelsIn a manner similar to that of hierarchical geometric models, hierarchical material models will beconstructed with a material distribution specified at the macro-scale and a different material modelspecified at a smaller scale. For cases where different materials are represented as separate phases atmicro or nano-scales, we can continue the example from Section 4.1. The nylon material with 10percent nano-filler would be represented as m

M= {0.9 m

1, 0.1 m

2}, where m

1= (1, 0) representing nylon

and m2

= (0. 1) representing nano-filler at the macro-scale. That is, at the macro-scale, the material isviewed as a continuum with a combination of materials 1 and 2. At the micro-scale, the differentmaterial phases are represented separately. The matrix material, m

1occupies the geometry of the part,

or the geometry of a subdivision, while the nano-filler has a rectangular geometry that is embedded inthe matrix.As an approximation, the hierarchical material model can represent material compositions in the partby interpolating between the macro-scale and micro-scale, or nano-scale, material models. Onesimplest approach is to progressively increase the content of each material phase in differentsubdivisions as the part model is progressively subdivided. The choice of which material phase toincrease in each subdivision should be made on a proportional basis. For the example of 90 percentnylon and 10 percent nano-filler, 9 out of 10 subdivision should favor nylon, while the 10thsubdivision has an increased percentage of nano-filler. Such intermediate models will have materialdistributions that approximate the desired distribution in the macro-scale and micro or nano-scalemodels.


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5. EXAMPLESThree examples will be used to illustrate the proposed hierarchical modeling method, one that usessimple geometry and material compositions, one that uses more complex micro-scale geometry, andone that includes a randomized fiber orientation to mimic that caused by injection molding.

5.1 Rectangular Plate with Particulate FillerThe first example has a simple rectangular shape 10x8 cm in size, with a particulate filler that isuniformly distributed throughout the part at a volume fraction of 0.1 (10% by volume). For simplicity,we will assume that the particles are roughly rectangular in shape about 2.5 mm in size. At eachsubdivision step, the geometry will be decomposed such that rectangles are divided into 4 equallysized rectangles. As such, 5 subdivision steps are needed to generate the “micro-scale” from themacro-scale. Linear interpolation of material compositions will be used. For visualization purposes,red will be used to indicate the polymer material that comprises the part, and blue will be used for theparticles. The initial material composition (90% “red” and 10% “blue”) corresponds to an initial colorthat is a reddish purple.

Using the notation from earlier, the macro-scale geometry, gM

M, is a rectangle, the micro-scale

geometric models {g1, g2

} (g1, g2

) are also rectangles, where g1represents the matrix material

m1

and g2represents the nano-filler m

2, the two materials are m

1= (1, 0), m

2= (0, 1), m

M

Mis (0.9m

1,

0.1 m2) and m is {m

1, m

2}. As a result, s

M= (g

M, m

M), s

M

M, s = {( g1

, m1), (g2

, m2)}, s , s

n=

, and rulesM

= {subdivision by mapping, relate size scales via linear interpolation}.Fig. 5 shows the results of the second, third, fourth, and fifth subdivision steps. Note that the red andblue colors become more pure until step 5 when the colors are pure red and pure blue. Every tenth cellin the model has its concentration of material 2 (blue) increased, while the other nine cells increase inmaterial 1. At the fifth step, every tenth cell will be 100 percent material 2, which is consistent withthe 10 percent nano-filler (material 2) specification. The material compositions in the reddish andbluish cells are given in Table 1 for the macro-scale part and for each subdivision step. Note that insubdivision step 1, only the geometry is divided. The material composition remains the same, sinceonly 4 cells were created, far less than the 10 cells necessary to start introducing the particles into thematerial composition.

5.2 Curved Plate with Cellular StructureThe second example illustrates the capability of dividing curved geometry, as well as the specificationof different geometries at different size scales. Again, the plate is about 10x8 cm in size, but withcurved boundaries. Also, the plate is planar and is modeled as a bicubic Bezier surface patch. Themicro-scale geometry is the hollow diamond shown in Fig. 4. As with the previous example, the micro-scale geometry is enlarged for purposes of illustration, so the diamond-shaped cells are 2.5 mm on aside, which results in the need for 5 subdivision steps. A mapped mesh approach is taken forsubdivision. At each subdivision step, each quadrilateral cell is divided into four smaller cells as in theprevious example. The material composition at the macro-scale is all one material, represented by ablue color. At the micro-scale, the hollow diamonds are all of the same material. The void regions arerepresented as a white color (no material).

Subdivision Step Matrix Cells Nano-Filler CellsProportion of

Material 1Proportion of



Material 20 0.8 0.2 - -1 0.8 0.2 - -2 0.85 0.15 0.15 0.853 0.9 0.1 0.1 0.94 0.95 0.05 0.05 0.955 1 0 0 1

Tab. 1: Material compositions for each subdivision step.


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Fig 5: Hierarchical material example.

At intermediate size scales, the material composition will be interpolated linearly between the materialcompositions at the macro-scale and the micro-scale. As a result, only the void regions (micro-scale)vary as the size scale is reduced.The third, fourth, and fifth subdivision steps are shown in Fig. 6a,b,c, respectively. Note that the “void”region becomes lighter as the size scale decreases, becoming white in the fifth step.

5.3 Randomized Fiber OrientationThe third example illustrates how microstructure can be varied as part geometry varies. In this case,the orientation and density of fiber filler will vary to mimic variations observed in injection moldedparts. In particular, fiber distributions become denser and aligned with the flow as polymer fills themold cavity. For this example, the microstructure geometry is a quadrilateral region with a 6 mm longfiber that has a random orientation and position within the quadrilateral. As the polymer flows intonarrowed regions, the fibers become oriented in the flow direction. As shown in Fig. 7, the polymerflow is from left to right; one can observe that fibers are randomly oriented in the wide region on theleft, but become progressively aligned horizontally as the narrower right side is reached. The fiber


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density is nominally about 10 percent, but becomes denser as one moves to the right. Fig. 7 illustratesthe 5th subdivision step.

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Fig. 6: Hierarchical geometry example.

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Fig. 7: Randomized fiber orientation example; 5th subdivision.


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6. CLOSUREHierarchical modeling in this paper refers to the modeling of parts with solid geometry, materialcomposition, and possibly distributions of properties, all at multiple length scales. A brief survey wasperformed on topics related to hierarchical modeling, specifically heterogeneous and hybrid modeling,subdivision methods, and level of detail approaches. The capabilities of these technologies werecompared against the requirements for modeling cellular materials and other structures with complexgeometric and material constructions. A specific hierarchical modeling method was proposed. Three2D examples illustrated the application of the proposed modeling method. From this preliminarywork, the following conjectures can be made:

new types of models are needed to represent objects with geometry, material, and propertydistributions at multiple length scales, since existing approaches to heterogeneous and hybridmodeling are insufficient;

the proposed approach that combines subdivision with material and/or property interpolationover the subdivided regions is effective in capturing the desired distributions at multiplelength scales;

two complementary approaches seem promising: the proposed subdivision approachpresented here and the implicit modeling approach, particularly those that are based onhypertextures;

since the examples are simple and only 2D, considerably more work is needed to more fullydevelop the proposed hierarchical modeling methods and explore its application to morecomplex part designs.

7. ACKNOWLEDGEMENTSWe gratefully acknowledge the U.S. National Science Foundation, through grant DMI-0522382, and thesupport of the Georgia Tech Rapid Prototyping and Manufacturing Institute member companies,particularly Pratt & Whitney and Paramount Industries, for sponsoring this work.

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